Question

Sibling IQ Scores: There have been numerous studies involving the correlation and differences in IQ's among siblings. Here we consider a small example of such a study. We will test the claim that, on average, older siblings have a higher IQ than their younger sibling. The results are depicted for a sample of 10 siblings in the table below. Test the claim at the 0.05 significance level. You may assume the sample of differences comes from a normally distributed population.

Pair ID	Older Sibling IQ (x)	Younger Sibling IQ(y)	difference (d = x − y)
1	86	84	2
2	89	94	-5
3	93	87	6
4	94	94	0
5	101	97	4
6	106	104	2
7	107	107	0
8	112	111	1
9	116	110	6
10	123	115	8
Mean	102.7	100.3	2.4
s	12.2	10.6	3.8

If you are using software, you should be able copy and paste the data directly into your software program.

(a) The claim is that the mean difference is positive (μ_d > 0). What type of test is this?

This is a two-tailed test.

This is a left-tailed test.   

This is a right-tailed test.

(b) What is the test statistic? Round your answer to 2 decimal places.
t_d =
To account for hand calculations -vs- software, your answer must be within 0.012 of the true answer.

(c) Use software to get the P-value of the test statistic. Round to 4 decimal places.
P-value =

(d) What is the conclusion regarding the null hypothesis?

reject H₀

fail to reject H₀    

(e) Choose the appropriate concluding statement.

The data supports the claim that, on average, older siblings have a higher IQ than younger siblings.

There is not enough data to support the claim that, on average, older siblings have a higher IQ than younger siblings.

We reject the claim that, on average, older siblings have a higher IQ than younger siblings

We have proven that, on average, older siblings have a higher IQ than younger siblings.

Answer 1

The statistic software output for this problem is:

Paired T hypothesis test: Hp = 1 - H2: Mean of the difference between Older Sibling IQ (x) and Younger sibling IQ(y) Ho: Pp = 0 HA: PD > 0 Hypothesis test results: Difference Mean Std. Err. DF T-Stat P-value Older Sibling IQ (x) - Younger sibling IQ(Y) 2.4 1.1944315 9 2.0093241 0.0377

This is a right-tailed test.

Test statistics = 2.01

P-value = 0.0377

P-value < 0.05

Reject the null hypothesis

The data supports the claim that, on average, older siblings have a higher IQ than younger siblings.